Download Before reading the descriptions of the board work try to figure out

Finding the Nth-Term in the Growing Squares Problem
Before reading the descriptions of the board work try to figure out what you can about each
teacher's approach.
Kevin
Kevin noticed that the second figure had 3 horizontal rows of 2 toothpicks and 3 vertical columns
of 2 toothpicks. That is two groups of 6, or 12 toothpicks total. Then he saw that the same was true
for Figure 3. There were 4 horizontal rows of 3 toothpicks and 4 vertical columns of 3 toothpicks,
or 2 groups of 12 toothpicks, or 24 total toothpicks.
He saw that the same was true for Figure 4. There were 5 horizontal rows of 4 toothpicks and 5
vertical columns of 4 toothpicks, or 2 groups of 20 toothpicks, or 40 total toothpicks.
Kevin (continued)
Given the Nth figure, Kevin realized he would have 2 groups of N(N+1) toothpicks.
Eric
2n2 + 2n is 4 times the formula for calculating the triangular numbers (n2 + n)/2
Jane
Jane noticed the 4th figure has 4 rows of 4 horizontal toothpicks (pink), and 4 rows
of 4 vertical toothpicks (green) or 2(4)2 toothpicks. Then she saw that there were 2
rows of 4 toothpicks remaining (grey), one making up the right side of the square
and the other making up the bottom. The same was true for the 5th figure and Jane
was able to come up with the same generalization as Eric, 2x2 + 2x.
Cynthia
Cynthia saw that the number of toothpicks followed the pattern of 4 times the
sequence of triangular numbers. So for example, since the 2nd triangular number is
3, the # of toothpicks in the 2nd figure is 4(3). Since the 3rd triangular number is 6,
the # of toothpicks in the 3rd figure would be 4(6).
As with Eric and Jane's formula, Cynthia's is 4 times the formula for calculating the
triangular numbers - (n2 + n)/2
4(n(n+1))
4(n2 +n)
2
2
Mark, Charlie, Ramon
After finding the
iterative rule for figuring
out the number of
squares and the number
of toothpicks in each
figure, we got interested
(a.k.a. Mark got fixated)
in the idea of visualizing
separating each square
in a figure and trying to
figure out the pattern in
how many toothpicks
are duplicated (and
need to be removed),
when the squares are
put back together. For
example, the 2nd figure
is made up of 4 squares.
If we separate those 4
squares, it would take 16
toothpicks to construct
them. But when we put
them back together, we need to remove 4, leaving the 2nd figure with only 12
toothpicks. If we separate the 9 squares in the 3rd figure, it would be 36 toothpicks.
But when we put them back together we need to subtract 12 toothpicks, leaving us
with 24. When time ran out we were trying to find the relationship between the
figure number and the number of toothpicks we had to subtracted when we
brought the squares back together.
Figure Number
(x)
4(Figure
Difference between
Figure
Number
4(Figure Number
Number
# of
Squared) Squared) and the actual #
Squared toothpicks
of toothpicks in the
x2
figure
4(x)2
1
1
4
4
0
1(0)
2
4
12
16
4
2(2)
3
9
24
36
12
3(4)
4
16
40
64
24
4(6)
5
25
60
100
40
5(8)
6
36
84
144
60
6(10)
4x2 - ___________
The generalization to the left should read:
n2 + ((n+1)2 - 1)
Charlie came up with the idea by looking at the
relationship between the number of squares
and the number of toothpicks. He noticed in
Fig. 1, there were 4 toothpicks and 1 square - a
difference of 3. He noticed that figure 2 had 12
toothpicks and 4 squares a difference of 8. He
saw that Fig 3 had 24 toothpicks and 9 squares
- a difference of 15. Then he realized the the
difference between the # of toothpicks and the
number of squares for each figure was one less
than the number of squares in the next figure.
So for example, the difference between the
number of toothpicks and the number of
squares in the 5th figure would be 35, because
that is one less than the square of the next
figure (62 = 36). Since we know the 5th figure
has 25 squares, we know it would have 25 +
(36-1) toothpicks, which it does... 60
toothpicks.
But in testing the generalization as it is written in the photo, I realized it is what
was missing from the generalization we were trying to build above when we
separated and then recombined the squares...
4x2 - (n2 + (n-1)2 - 1),
Jane's Growing Triangles
In the growing triangles pattern (Fig 4 is the one drawn above), Jane noticed that
the same thing that happens to the vertical toothpicks, happens to the toothpicks
on both diagonals - three times in all. That is to say the outside edge of each
triangle is the figure number, and then the number of toothpicks goes down by one,
all the way to one toothpick. For example - in the 4th figure above, the right side of
the triangle has 4 black toothpicks, followed by 3, then 2, then 1. The same is true
for the sea foam green and the fern green. This was a pattern she recognized from
Gauss and the Handshake problem.
This formula is 3 times the formula for calculating triangular numbers: (n2 +n)/2